Question Video: Finding the Modulus of Complex Numbers Using Conjugates

Given that 𝑍 = (3 − 9𝑖)/(1 + 3𝑖), determine |𝑍|.

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Video Transcript

Given that 𝑍 is equal to three minus nine 𝑖 divided by one plus three 𝑖, determine the absolute value or modulus of 𝑍.

We recall that for any complex number of the form 𝑎 plus 𝑏𝑖, the modulus of that complex number is equal to the square root of 𝑎 squared plus 𝑏 squared. In order to divide any two complex numbers, we need to multiply the numerator and denominator by the complex conjugate of the denominator, as this creates a real number on the denominator. If 𝑍 is equal to 𝑎 plus 𝑏𝑖, then the complex conjugate 𝑍 star is equal to 𝑎 minus 𝑏𝑖. The complex conjugate of one plus three 𝑖 is one minus three 𝑖. Therefore, we need to multiply three minus nine 𝑖 over one plus three 𝑖 by one minus three 𝑖 over one minus three 𝑖.

We will do this by distributing the parentheses or expanding the brackets using the FOIL method. Multiplying the numerators gives us three minus nine 𝑖 minus nine 𝑖 plus 27𝑖 squared. We recall that 𝑖 squared is equal to negative one. As 27 multiplied by negative one is negative 27, the numerator simplifies to negative 24 minus 18𝑖. Distributing the parentheses on the denominators gives us one minus three 𝑖 plus three 𝑖 minus nine 𝑖 squared. Once again, we can replace 𝑖 squared with negative one. Negative three 𝑖 plus three 𝑖 is equal to zero. Therefore, the denominator simplifies to 10.

We can then split the real and imaginary parts. So we have negative 24 over 10 minus 18 over 10 𝑖. Both of these fractions simplify so that the complex number 𝑍 is equal to negative 12 over five minus nine over five 𝑖. We can now calculate the modulus of this complex number. We need to find the sum of the squares of the real and imaginary components and then square root our answer. This simplifies to the square root of 144 over 25 plus 81 over 25. Adding the numerators of our two fractions gives us 225. As 225 divided by 25 is equal to nine, the modulus of the complex number 𝑍 is equal to the square root of nine. This gives us a final answer of three.

If 𝑍 is equal to three minus nine 𝑖 divided by one plus three 𝑖, then the modulus of 𝑍 is equal to three.

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